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<!DOCTYPE html>

<head>

<title>Greedy Algorithm</title>

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<h3>Page Index</h3>

<ul>

<li><a href="/dsa.html#data-structure-1" title="Data Structures (I)">Data Structures

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<li><a href="/dsa.html#data-structure-2" title="Data Structures (II)">Data Structures

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<li><a href="/dsa.html#sorting-searching" title="Sorting and Searching">Sorting and

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<li><a href="/dsa.html#greedy-algorithm" title="Greedy Algorithms">Greedy Algorithms</a>

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<li><a href="/dsa.html#dynamic-programming" title="Dynamic Programming">Dynamic

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<h1>Greedy Algorithm</h1>

<p class="editor-contents__short-description">In this tutorial, you will learn what a Greedy Algorithm is. Also, you will find an example of a greedy approach.</p>

<p id="definition">A greedy algorithm is an approach for solving a problem by selecting the best option available at the moment. It doesn't worry whether the current best result will bring the overall optimal result.</p>

<p>The algorithm never reverses the earlier decision even if the choice is wrong. It works in a top-down approach.</p>

<p>This algorithm may not produce the best result for all the problems. It's because it always goes for the local best choice to produce the global best result.</p>

<p>However, we can determine if the algorithm can be used with any problem if the problem has the following properties:</p>

<p><strong>1. Greedy Choice Property</strong></p>

<p>If an optimal solution to the problem can be found by choosing the best choice at each step without reconsidering the previous steps once chosen, the problem can be solved using a greedy approach. This property is called greedy choice property.</p>

<p><strong>2. Optimal Substructure</strong></p>

<p>If the optimal overall solution to the problem corresponds to the optimal solution to its subproblems, then the problem can be solved using a greedy approach. This property is called optimal substructure.</p>

<hr><h2 id="advantages">Advantages of Greedy Approach</h2>

<ul><li>The algorithm is <strong>easier to describe</strong>.</li>

<li>This algorithm can <strong>perform better</strong> than other algorithms (but, not in all cases).</li>

</ul><hr><h2 id="drawback">Drawback of Greedy Approach</h2>

<p>As mentioned earlier, the greedy algorithm doesn't always produce the optimal solution. This is the major disadvantage of the algorithm</p>

<p>For example, suppose we want to find the longest path in the graph below from root to leaf. Let's use the greedy algorithm here.</p>

<figure><img alt="Apply greedy approach to this tree to find the longest route" src="https://www.programiz.com/sites/tutorial2program/files/greedy_approach_na.png" title="Apply greedy approach to this tree to find the longest route" width="460" height="292"><figcaption>Apply greedy approach to this tree to find the longest route</figcaption></figure><p></p><div class="clearfix"></div><p><strong>Greedy Approach</strong></p>

<p>1. Let's start with the root node <strong>20</strong>. The weight of the right child is <strong>3</strong> and the weight of the left child is <strong>2</strong>.</p>

<p>2. Our problem is to find the largest path. And, the optimal solution at the moment is <strong>3</strong>. So, the greedy algorithm will choose <strong>3</strong>.</p>

<p>3. Finally the weight of an only child of <strong>3</strong> is <strong>1</strong>. This gives us our final result <code>20 + 3 + 1 = 24</code>.</p>

<p>However, it is not the optimal solution. There is another path that carries more weight (<code>20 + 2 + 10 = 32</code>) as shown in the image below.</p>

<figure><img alt="Longest path" src="https://www.programiz.com/sites/tutorial2program/files/greedy_approach_longest.png" title="Longest path" width="460" height="292"><figcaption>Longest path</figcaption></figure><p>Therefore, greedy algorithms do not always give an optimal/feasible solution.</p>

<hr><h2 id="algorithm">Greedy Algorithm</h2>

<ol><li>To begin with, the solution set (containing answers) is empty.</li>

<li>At each step, an item is added to the solution set until a solution is reached.</li>

<li>If the solution set is feasible, the current item is kept.</li>

<li>Else, the item is rejected and never considered again.</li>

</ol><p>Let's now use this algorithm to solve a problem.</p>

<hr><h2 id="example">Example - Greedy Approach</h2>

<pre>Problem: You have to make a change of an amount using the smallest possible number of coins.

Amount: $18

Available coins are

$5 coin

$2 coin

$1 coin

There is no limit to the number of each coin you can use.</pre>

<p><strong>Solution:</strong></p>

<ol><li>Create an empty <code>solution-set = { }</code>. Available coins are <code>{5, 2, 1}</code>.</li>

<li>We are supposed to find the <code>sum = 18</code>. Let's start with <code>sum = 0</code>.</li>

<li>Always select the coin with the largest value (i.e. 5) until the <code>sum > 18</code>. (When we select the largest value at each step, we hope to reach the destination faster. This concept is called <strong>greedy choice property</strong>.)</li>

<li>In the first iteration, <code>solution-set = {5}</code> and <code>sum = 5</code>.</li>

<li>In the second iteration, <code>solution-set = {5, 5}</code> and <code>sum = 10</code>.</li>

<li>In the third iteration, <code>solution-set = {5, 5, 5}</code> and <code>sum = 15</code>.</li>

<li>In the fourth iteration, <code>solution-set = {5, 5, 5, 2}</code> and <code>sum = 17</code>. (We cannot select 5 here because if we do so, <code>sum = 20</code> which is greater than 18. So, we select the 2nd largest item which is 2.)</li>

<li>Similarly, in the fifth iteration, select 1. Now <code>sum = 18</code> and <code>solution-set = {5, 5, 5, 2, 1}</code>.</li>

</ol><hr><h2 id="applications">Different Types of Greedy Algorithm</h2>

<ul><li><a href="/dsa/selection-sort.html">Selection Sort</a></li>

<li><a href="https://en.wikipedia.org/wiki/Knapsack_problem" target="_blank">Knapsack Problem</a></li>

<li><a href="/dsa/spanning-tree-and-minimum-spanning-tree.html">Minimum Spanning Tree</a></li>

<li><a href="https://en.wikipedia.org/wiki/Shortest_path_problem" target="_blank">Single-Source Shortest Path Problem</a></li>

<li>Job Scheduling Problem</li>

<li><a href="/dsa/prim-algorithm.html">Prim's Minimal Spanning Tree Algorithm</a></li>

<li><a href="/dsa/kruskal-algorithm.html">Kruskal's Minimal Spanning Tree Algorithm</a></li>

<li><a href="/dsa/dijkstra-algorithm.html">Dijkstra's Minimal Spanning Tree Algorithm</a></li>

<li><a href="/dsa/huffman-coding.html">Huffman Coding</a></li>

<li><a href="/dsa/ford-fulkerson-algorithm.html">Ford-Fulkerson Algorithm</a></li>

</ul></div>

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<div class="tutorial-toc"><div class="tutorial-toc__inner"><h3 class="tutorial-toc__title">Table of Contents

<li><a href="#advantages">Advantages of Greedy Approach</a></li>

<li><a href="#drawback">Drawback of Greedy Approach</a></li>

<li><a href="#algorithm">Greedy Algorithm</a></li>

<li><a href="#example">Example - Greedy Approach</a></li>

<li><a href="#applications">Different Types of Greedy Algorithm</a></li>

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document.getElementById('cpp1').className = 'tabbed-editor__node';

document.getElementById('python-code').className = 'code-editor__area';

document.getElementById('java-code').className = 'code-editor__area';

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document.getElementById('cpp-code').className = 'code-editor__area';

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